Why is my model distorting when I weight paint?
This is because you have moved some of the bones while in pose mode. While weight painting you are changing how the vertices of the mesh are being influenced by the bones.

To fix this, select the armature and go to pose mode. Then select every bone and press alt+g and alt+r to reset the location and rotation of each bone respectively. (alt+s to reset scale)

Then with your mesh selected and in weight paint mode, you shouldn't see any more deforms while painting because the armature is in its default position.